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1 TMA420, Matematikk 4K, Fall 206 LECTURE SCHEDULE AND ASSIGNMENTS Date Section Topic HW Textbook problems Suppl Answers Aug 22 6 Laplace transform 6:,7,2,2,22,23,25,26,4 A Sept 5 Aug 24/ ODE, Heaviside function 62:9,27; 63:5,3,40 Dirac delta Aug :5,4abc;65:7,3,9,6e B Convolution 2 Sept 2 Aug 3/ :6,7;67:3; 6R:39 C Systems of ODE Sept 5 Sept 7/8 Sept 2 Sept 4/5 Sept 9 Sept 2/22 Sept 26 Sept 28/29 Oct 3 Oct 5/6 Oct 0 Oct 2/3 Oct 7 Oct 9/20 Oct 24 Oct 26/27 Oct 3 Nov 2/3 Nov 7 Nov 9/0 Nov 4 Nov 6/7 Nov 2 Nov 23/24 Dec 3 2, 3-2 4* 4, , (5) Fourier Series Complex numbers Complex Fourier series Approximation Forced oscillations Fourier Integral Fourier Transform PDE, Wave Equation Heat Equation Heat and wave eqn Analytic functions Cauchy-Riemann eqns Exponential Trig fns, Log Conformal mappings Complex integration Cauchy Theorem Cauchy formula Derivatives Power Series Functions by Power Series Taylor and Maclaurin Singularities and zeros Residue Integration Real integrals Applications Repetition - Final Exam 3 4 :9,4,9,2;2:,7,25 3:9,2,8,9;32:3,8,2 3:5,9;4*:9,0,3 4:4,8,;:R:5,7 D Sept 9 E,F,G Sept :2,,9;9:4,9 H,I,J K,L,M Oct 3 6 2:3,9,5;23:,7,5,6,7 N 26:5,2 O Oct :3;27:2,3 P 33:3,7,8,0,,4,2,23 Q Oct :3,9,3;35:5,6 36:3,9,8; 37:7,7,9,23 R S Oct 24 7:5,8,,3,5 Oct 3 4:4,7,2,7,2,23,35;42:4,22,27 43:3,2,8;44:3,4,8,5 5:,2,6,7,9,30;52:5,6,9 53::4,7,6;54:5,8,9,9,24 5R:4,8,26,29 6:3,7,3;62:3,5,6a,7 T 63:,6,9;64:3,6 U V,W,X,Y Z,Æ,Ø,Å 4* refers to 4 from 9th edition (a scan of the pages can be found on the web-page) 78: Consider only the lines x = 2 and y = 3 75: Take the cubic polynomial z 3 + 3z : Find all zeros and poles of the given function and determine their orders Nov 7 Nov 4 Nov 2 Nov 24

2 Supplementary problems A Let y(t) be the solution of the differential equation y (t) + y (t) 2y(t) = r(t) that satisfies y(0) =, y (0) =, where the function r(t) is given by its graph: y x Find the Laplace transform of y B Use The Laplace transform to solve the differential equation y + 4y + 4y = 2e 2t + δ(t ), t > 0 with initial conditions y(0) = 0 and y (0) = 0, where δ is the Delta function C Use The Laplace transform to solve the initial value problem y + y + t 0 y(τ)e t τ dτ = u(t ), t > 0, y(0) = D The function f is defined by the following conditions: i) f(x) = f( x) for all real x ii) f(x) = f(x + 4) for all real x iii) f(x) = x for 0 < x < 2 Sketch the graph of f for 2 < x < 6 Find the Fourier series of f E Let f(x) = x(π x) for 0 x π The Fourier Sine series of f is given 8 sin(2m + )x π (2m + ) 3 Determine the sum of the series F The 2π-periodic function f is defined by f(x) = e x, π < x < π a) Sketch the graph of the periodic extension f and find the complex Fourier series of f b) Determine the sums of the series: n=2 m=0 ( ) n + n, 2 n=2 + n 2

3 G Let f be the 2π-periodic function defined by f(x) = x 4 for π < x π The Fourier series of f is π n= ( ) n 8(π 2 n 2 6) n 4 Use this to find the sums of the series: π 2 n 2 6 (i), (ii) n 4 n= n= cos(nx) π 4 n 4 2π 2 n n 8 H Compute the Fourier transform of the function e x, x 0 f(x) = e x, x < 0 Use the result to compute the integral 0 w sin w + w 2 dw I Compute the Fourier transform of the function f(t) = cos(t)e t2 J Find the Fourier transform of the function h(x) = e x2 e x2 Use the result to express h(x) without an integral or convolution sign K Use the Fourier transform to solve the equation f(x) e 3 x t f(t)dt = e 3 x L Functions f(x) and g(x) are defined by for x < f(x) = 0 otherwise a) Show that the Fourier transforms of f and g are, g(x) = e x for x > 0 0 otherwise ˆf(w) = 2 sin w π w and ĝ(w) = 2π iw + w 2 b) Let h(x) be the convolution of f and g, h(x) = (f g)(x) = f(x y)g(y)dy Prove that and determine the value of the integral h(x) = π ( iw) sin w e iwx dw w( + w 2 ) sin w w( + w 2 ) dw

4 M Let f(x) =, x < 0, x Show that the Fourier transform of f can be written as ˆf(w) = i e iw e iw 2π w Find also the Fourier transform of the convolution f f Use the inverse Fourier transform to determine the value of the integral cos 5w 2 cos 3w + cos w dw (You can use without proof that f f is a continuous function) N A complex number z 0 satisfies e z 0 = 5 Find the value of e 2z 0+3i O Let a and b be two real constants, consider two boundary value problems 2 u = 0, for t > 0, x (0, ) t x 2 ( ) u(0, t) = a fort > 0, u(, t) = b fort > 0, and ( ) w 2 2 u = 0, for t > 0, x (0, ) t x 2 u(0, t) = 0 fort > 0, u(, t) = 0 fort > 0, a) Let u and u 2 be solutions of (*) Determine which boundary value problem are solved by functions u + u 2 and u u 2 Does the superposition principle hold for (*) and/or for (**)? b) Let u(x, t) be a solution of the boundary value problem (*) and let v(x, t) be defined by v(x, t) = u(x, t) (a + (b a)x), for t 0 and x [0, ] Show that v(x, t) is a solution of (**) Find all solutions of (**) of the form v(x, t) = F (x)g(t) c) Let a = and b = in (*) Find a solution of the boundary value problem (*) that satisfies the initial condition u(x, 0) = sin(πx) for 0 < x < P a) Find the Fourier Sine Series of the function f(x) = πx x 2, 0 x π b) Let u(x, t) be a solution of the boundary value problem u t = u xx 2u x, 0 < x < π, t > 0, u(0, t) = 0 = u(π, t), t > 0 Show that if u(x, t) = F (x)g(t) then F (x) = Ce x sin nx for some integer n c) Find a solution u(x, t) of the boundary value problem in b) such that u(x, 0) = e x f(x), 0 < x < π, where f(x) is the function given in a) Q Determine which of the following functions are analytic at z 0 = (i) zre(z), (ii) z 2, (iii) z R Find all solutions to the equation e 2z = i S The function f(z) = y 3 + Bx 2 y + iv(x, y) is analytic Determine the constant B and the function v(x, y) if v(0, 0) = 0 (Hint: Use the Laplaces and Cauchy Riemann equations) T Find all Laurent series with center z = of the function f(z) = z + ez z

5 U a) Find all Laurent series with center z = 0 of the function f(z) = z(8z 3 ) and determine the domain of convergence for each series b) Let C be the unit circle z = with positive orientation (counter clockwise) Determine the values of the integrals f(z)dz and (Rez)dz V Solve the equation C y (t) + y(t) = with initial conditions y(0) = 0, y (0) = 0 2 sin 2t for 0 t π 0 for t > π, W Let f(x) be the 2π-periodic function defined by the graph: f(x) 2π C π 4π 3π 2π π π 2π 3π 4π x π a) Determine the value of the sum of the Fourier series of f at points x = 0 and x = π b) Find the Fourier series of f X The temperature in an isolated bar satisfies the equation ( ) t = 2 u + 2u, 0 < x < π, t > 0 x2 with boundary conditions ( ) (0, t) = 0 = (π, t) x x (The term 2u appears because only the ends of the bar are isolated) a) Find all solutions of the boundary value problem ( ), ( ) of the form u(x, t) = X(x)T (t) b) Use the superposition principle to find the solution satisfying the initial condition u(x, 0) = (cos(x) + ) 2, 0 < x < π Y Determine all values of c such that the function u(x, y) = e cx sin y cos y is harmonic Find all analytic functions f(z) such that Re(f(z)) = u(x, y) when z = x + iy Z Expand the function f(z) = e /z2 in A Laurant series that converges for 0 < z < R and find its region of z 2 convergence Compute the residue Res z=0 f(z)}

6 Æ It is given that e 2ix x 2 + 6x + 25 dx = 2πi } e 2iz Res, z 2 + 6z + 25 where the sum is taken over the singular points of the function in the upper half-plane Compute the integral and determine the value of cos 2x x 2 + 6x + 25 dx Ø Use the residue calculus to compute the integral 2π dθ sin θ Å a) Show that the function f(z) = eiz e 5iz z 2 0 can be written as f(z) = b z + g(z) for z > 0, where b is a complex number and g(z) is an analytic function Determine the value of b b) Let S R be the half-circle parametrized by z = Re iθ, 0 θ π Show that f(z)dz 4π when R 0 S R c) Show that f(z)dz 0 when R S R and compute the integral cos x cos 5x dx 0 2x 2 (You may assume that the last integral converges without proving it)

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